Nobody actually uses undergraduate style vectors in real life, they are an inane useless outdated formalism which should not be taught.
What people do use is the mathematically less sophisticated, but practically more useful, decomposition of vectors into components. The undergraduate calculus of cross products and dot products is incomplete, because it excludes operations which produce symmetric tensors, which show up all the time, and it is unweildy, because the cross product identities are counterintuitive.
The real life formalism everybody uses is tensor index notation, as used and developed by Einstein and others at the turn of the 20th century. This notation replaces vector notation, is universal for tensors, and is directly reducible to component computations. When learning elementary physics, it is best to translate everything to index notation as quickly as possible.
The history of vectors is William Rowan Hamilton's introduction of quaternions. Quaternions had a dot-product/cross-product multiplication, but they had 4 components. Physicists liked quaternions because they were mathematically elegant, and many papers used quanternions to express physical quantities. But in the 20th century, it became increasingly clear that quaternions were a peculiar algebraic structure which were useful for Lie groups, but had nothing to do with our three-dimensional space or our four dimensional spacetime. So physicists extracted the dot and cross product from the quaternion formalism, which butchered the whole scheme. The quaternions are a division algebra. Vectors with cross products are a nothing algebra. The quaternions are associative. Vectors under cross products aren't. All the elegance of quaternions was gone, and the clunkiness of the ill-fitting notation remained.
